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Chapter 6: Problem 74

If two matrices can be multiplied, describe how to determine the order of the product.

Short Answer

Expert verified
The order of the product of two matrices is determined by the number of rows from the first matrix and the number of columns from the second matrix. If the first matrix has dimensions m x n and the second matrix has dimensions p x q, for multiplication to be possible, n must be equal to p. However, the order of the product matrix will be m x q.

Step by step solution

01

Understand Matrices Dimension

To start, one needs to understand the dimensions of matrices. A matrix has rows and columns. The dimension of a matrix is given as (Number of Rows) x (Number of Columns).
02

Conditions for Multiplication

Multiplication of two matrices is possible only when the number of columns in the first matrix is equal to the number of rows in the second matrix. If Matrix A has dimensions m x n, and Matrix B has dimensions p x q, then for multiplication to be possible, n (the number of columns in Matrix A) must be equal to p (the number of rows in Matrix B).
03

Determine the Order of the Product

Once multiplication is verified to be possible, the order of the product matrix (Matrix C) is determined by the number of rows from the first matrix (m) and the number of columns from the second matrix (q), resulting into a matrix of order m x q.

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Most popular questions from this chapter

What is the multiplicative identity matrix?

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows If an operation is not defined, state the reason. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$A-C$$

What is a cryptogram?

Find \((A B)^{-1}, A^{-1} B^{-1}\), and \(B^{-1} A^{-1}\). What do you observe? $$A=\left[\begin{array}{ll}2 & -9 \\\1 & -4\end{array}\right] \quad B=\left[\begin{array}{ll}9 & 5 \\ 7 & 4\end{array}\right]$$

Use a graphing utility to evaluate the determinant for the given matrix. $$\left[\begin{array}{rrrrr}8 & 2 & 6 & -1 & 0 \\\2 & 0 & -3 & 4 & 7 \\\2 & 1 & -3 & 6 & -5 \\\\-1 & 2 & 1 & 5 & -1 \\\4 & 5 & -2 & 3 & -8\end{array}\right]$$
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